Derivation of Bernoulli's Principle

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Discussion Overview

The discussion revolves around the derivation of Bernoulli's principle in fluid mechanics, specifically comparing different methods of derivation, including the Euler equation and energy conservation principles. Participants explore the theoretical foundations and implications of these approaches.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that their Fluid Mechanics textbook uses the Euler equation for deriving Bernoulli's principle, while another textbook employs energy conservation, questioning the rationale behind the choice of method.
  • Another participant asserts that the Navier-Stokes equations are fundamental in fluid mechanics and that Bernoulli's equation is a special case derived from these equations, indicating that it encompasses various fluid conditions.
  • It is mentioned that Bernoulli's principle can be derived from the full Navier-Stokes equations by ignoring viscosity, leading to the Euler equations, which are based on conservation laws.
  • A participant expresses a desire to learn more about the cross-product rules, indicating a struggle with the mathematical aspects involved in these derivations.

Areas of Agreement / Disagreement

Participants present multiple competing views regarding the derivation methods for Bernoulli's principle, with no consensus reached on the preferred approach or its implications.

Contextual Notes

Participants express varying levels of comfort with the mathematical requirements of the derivations, which may influence their understanding and preference for different methods.

MaxManus
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In my Fluid Mechanics textbook they use the Euler equation to derive Bernoulli's principle for incompressible fluids with gravity. In my earlier introduction to mechanics textbook they used energy conservation. Is there a reason for using the Euler equation or is it just to show more ways to derive Bernoulli's principle.
The reason I ask is because I find the Euler method more difficult.

I use Kundu/Cohen Fluid Mechanics
 
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well the origin in fluid mechanics theoris is NAVIER STOKES EQUATIONS as they cover all the conditions of the fluid as ( steady or unsteady ) comprissible or not , newtenion or not and bernollis equation is a special case of the navier stokes and so is eulers you can study this equation it will enable you to study many parameters pressure , time , velocity k, flow rate even power also
 
As maxx mentioned Bernoulli's is derived from the full Navier-Stokes equations. Taking the full set of equations and ignoring viscosity you get the Euler equations.

Recall that these governing equations are based on Conservation of Mass, Momentum, and Energy. Those form a set of 5 coupled equations.

Bernoulli's is a special case basically of the energy equation. So, it is derived from Euler equation which is derived from conservation of energy.

edit: Perhaps you can show us the two different approaches. We might be able to help a little more knowing what you're looking at.
 
Thanks, then I will sit down and try to learn the cross-product rules.

Minger: the problem is my math knowledge.
 

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